The out-of-time-ordered correlation (OTOC) function is an important new probe in quantum field theory which is treated as a significant measure of random quantum correlations. In this paper, using for the first time the slogan “Cosmology meets Condensed Matter Physics”, we demonstrate a formalism to compute the Cosmological OTOC during the stochastic particle production during inflation and reheating following the canonical quantization technique. In this computation, two dynamical time scales are involved—out of them, at one time scale, the cosmological perturbation variable, and for the other, the canonically conjugate momentum, is defined, which is the strict requirement to define the time scale-separated quantum operators for OTOC and is perfectly consistent with the general definition of OTOC. Most importantly, using the present formalism, not only one can study the quantum correlation during stochastic inflation and reheating, but can also study quantum correlation for any random events in Cosmology. We have studied the possibility of having three different types of correlators, which quantifies the random quantum correlation function out-of-equilibrium. We have also studied the classical limit of the OTOC and checked the consistency with the large time limiting behaviour of the correlation. Finally, we prove that the normalized version of OTOC is completely independent of the choice of the preferred definition of the cosmological perturbation variable.

Distance-based topological invariants, namely the topological roundness index and the Wiener number, show a new and somehow unexpected symmetry between toroidal and Klein bottle polyhexes. In this case the bottles are closed in the anti-parallel way along the zigzag edge of the hexagonal lattice. We report here that our computations point out that both cubic graphs are topologically indistinguishable for certain combination of x,y sizes. This means that an Escher's ant walking on the Klein bottle is no longer able to distinguish it from a same-size Torus by measuring the chemical distances of a node from all the others. Among other effects, this new topological similarity does transfer the translation invariance, that is a typical feature of the graphenic Tori, to the Klein bottle lattices. This size-induced phase transition connecting Klein bottles and toroidal cubic graphs represents a relevant topological behavior with uncharted mathematical and physical consequences. The non-trivial influence of the chirality of the bottle will be also numerically investigated showing a radically different behavior of the armchair Klein Bottles.

One of the main questions of fundamental physics is the problem of the asymmetry matter/antimatter in the universe and the action of gravity on antimatter. Tests on antimatter gravity have currently a limited precision, with the sign of gravity acceleration not yet known experimentally. Ambitious projects are developed at CERN facilities to produce low energy antihydrogen with the aim of measuring the free fall of antihydrogen atoms. Among them, the GBAR experiment (Gravitational Behaviour of Antihydrogen at Rest) aims at measuring the gravity acceleration of antihydrogen atoms during a free fall in Earth’s gravitational field. The simulation of the free-fall chamber includes the Monte-Carlo generation of trajectories and the statistical analysis. A precision of the measurement beyond the % level is confirmed by taking into account the experimental design. We also propose a new method using quantum reflection of antiatoms above a reflecting mirror followed by a classical free fall; the quantum interference pattern obtained at detection improves the accuracy of the experiment by approximately 3 orders of magnitude.

The presence of symmetry and periodicity in optimization algorithms can have either positive or negative influence on dynamics of decision making. For instance, consider a cycling effect while solving the linear programming problem by the simplex method. The cycling is harmful and may cause a simplex method to get stuck in a local optimum; another example is the sub-cycles that are prohibited when solving the traveling salesman problem. On the other hand, symmetry and periodicity can be very useful when they help to quickly reach the optimum for robotic scheduling problems; such is the case, also, when the shortest path in a graph is sought bidirectionally from a start node and concurrently from an end node. The first part of the paper describes common types of symmetry and their effects on efficiency of the scheduling algorithms. In the second part, we present typical examples of periodic algorithms in scheduling theory. Finally, we present a new asymptotically periodic algorithm for solving a discrete search problem under uncertainty, when the overlooking and false alarms may occur at each step of the algorithm.

The gas dynamics equations with the state equation of a general form have the following symmetries: space translations, time translation, rotations, Galilean translations, uniform dilatation. In this investigation the state equation is a pressure equal to the sum of two functions - the first function depends on density, and the second function depends on entropy [1]. Such system of equations has additional symmetry – pressure translation. The system admits a 12-dimensional Lie algebra. An optimal system of dissimilar subalgebras of the 12-dimensional Lie algebra was constructed in [2].

Invariant submodels of rank 1 are calculated for 3-dimensional subalgebras [2]. The submodels are the systems of ordinary differential equations. Exact solutions were found for some submodels [3]. The motion of particles and volumes according to the exact solution is considered.

The author was supported by the Russian Foundation for Basic Research (project no. 18-29-10071) and partially from the Federal Budget by the State Target (project no. 0246-2019-0052).

References

[1] Ovsyannikov L.V. The “podmodeli” program. Gas dynamics // Journal of Applied Mathematics and Mechanics, 1994, vol. 58, no. 4, pp. 601-627. doi:10.1016/0021-8928(94)90137-6

[2] Siraeva D.T. Optimal system of non-similar subalgebras of sum of two ideals // Ufa Mathematical Journal. 2014. Vol. 6, No 1. Pp.90-103. Doi:10.13108/2014-6-1-90

[3] Siraeva D.T. Two invariant submodels of rank 1 of the hydrodynamic type equations and exact solutions // Journal of Physics: Conference Series, 1666 012049 (2020). Doi: 10.1088/1742-6596/1666/1/012049

The system of 16-component equations including two equations of
the Bethe-Salpeter kind (without an interaction) and two
additional conditions are considered. It is shown that the group
of the initial symmetry is $SU(3)_{C}\times SU(2)_{L}\times
U(1)$. The symmetry group is established as the consequence of the
field equations; $ SU(2)$ should be chiral, the color space has
the signature $ (++-)$. The structure of permissible multiplets of
the group coincides with the one postulated in the
$SU(3)_{C}\times SU(2)_{L}$-model of strong and electroweak
interactions excluding the possible existence of the additional $SU(2)_{R}
$-singlet in a generation. It is shown here that at least three puzzling features of the
standard model: the existence of a few generations, the specific
symmetry group, and the necessity to use its interwoven
representations may originate from the composite nature of the
fundamental fermions. \footnote{This paper (in Russian) was deposited in VINITY
19.12.1988 as VINITI No 8842-B88; it was an important stage in the
development of my model of the composite fundamental fermions (see
hep-th/0207210). Now I have translated it in English (small
corrections are made) to do more available.}

Symmetric categories have been of great interest in quantum algebra and mathematical physics. Cohen and Westreich in 1998 studied symmetries in the Yetter-Drinfel'd category over a Hopf algebra under some conditions. Pareigis in 2001 found the necessary and sufficient condition for $\!^{H}_{H}\mathcal{YD}$ to be symmetric. Later, Panaite et al. in 2010 proposed the definition of pseudosymmetric braided categories which can be viewed as a kind of weakened symmetric braided categories, and showed that the category $\!_{H}\mathcal{YD}^{H}$ is pseudosymmetric if and only if is commutative and cocommutative. Let $H$ be a Hopf algebra and $\mathcal{LR}(H)$ the category of Yetter-Drinfel'd-Long bimodules over $H$. We first show that the Yetter-Drinfel'd-Long category $\mathcal{LR}(H)$ is symmetric if and only if $H$ is trivial in four different methods, and that $\mathcal{LR}(H)$ is pseudosymmetric if and only if $H$ is commutative and cocommutative. We then introduce the definition of the $u$-condition in $\mathcal{LR}(H)$ and give a necessary and sufficient condition for $H_{i}$ $(i=1,2,3,4)$ to satisfy the $u$-condition. Then we study the relation between the $u$-condition and the symmetry of $\mathcal{LR}(H)$.

For a singular hypersurface of arbitrary type in quadratic gravity motion equations were obtained using only the least action principle. The coefficients in the motion equations are zeroed with a combination corresponding to the Gauss-Bonnet term. Therefore it does not create neither double layers nor thin shells. It has been demonstrated that there is no “external pressure” for any type of null singular hypersurface. For spherically symmetric lightlike singular hypersurfaces additionally the “external flux” is equal to zero and the system of motion equations is reduced to one which is expressed through the invariants of spherical geometry along with the Lichnerowicz conditions. In this case there are no double layers only thin shells. Spherically symmetric null thin shells were investigated for spherically symmetric solutions of conformal gravity as an application, in particular, for various vacua and Vaidya-type solutions. By virtue of Lichnerowicz conditions scalar curvature R₂ of the two-dimensional non-spherical part of the metric without conformal factor must be continuous on the null shell. Therefore the following combinations for two vacua are possible: matching a vacuum with a constant R₂ and a vacuum with a variable R₂, matching two vacuums with a variable R₂, matching two vacua with a coinciding constant R₂. In the first case, the hypersurface is an analogue of the double horizon for the vacuum with variable R₂; in other cases, junction is possible only if the metrics coincide up to a conformal factor. With the addition of the Vaidya-type solution new possible matchings appear: Vaidya-type metric with vacuum with variable R₂ and two Vaidya-type metrics. In the first version, the null shell is actually the singular part of the Vaidya-type solution, in the second, they must coincide up to the conformal factor. Moreover the null shell does not emit in both cases.

A new symmetry for Newtonian Dynamics is analyzed, this corresponds to going to an accelerated frame, which introduces a constant gravitational field into the system and subsequently we consider the addition of a linear contribution to the gravitational potential $\phi$ which can be used to cancel the gravitational field induced by going to the accelerated from, the combination of these two operations produces then a symmetry. This symmetry leads then to a Noether current which is conserved. The Conserved charges are analyzed in special cases. The charges may not be conserved if the Noether current produces flux at infinity, but such flux can be eliminated by going to the CM system in the case of an isolated system. In the CM frame the Noether charge vanishes,
Then we study connection between the Cosmological Principle and the Newtonian Dynamics which was formulated via a symmetry of this type, but without an action formulation. Homogeneous behavior for the coordinate system relevant to cosmology leads to a zero Noether current and the requirement of the Newtonian potential to be invariant under the symmetry in this case yields the Friedmann equations, which appear as a consistency condition for the symmetry.

In this work, we study a generalization of the well-known Kelvin-Voigt viscoelasticity equation describing the mechanical behavior of viscoelasticity. We perform a Lie symmetry analysis. Hence, we obtain a classification of the Lie point symmetries of the equation. Afterward, it is important to classify invariant solutions according to the classification of the associated symmetry generators, so we find the set of exactly one generator from each class. This problem of obtaining an optimal system of subgroups is equivalent to that of obtaining an optimal system of subalgebras. For this classification problem, we use the adjoint representation. Then, we transform the partial differential equation into an ordinary differential equation, by using the symmetry reductions of each element of the optimal system. Furthermore, we determine the conservation laws of this equation, by applying the multiplier method, developed by Anco and Bluman. Finally, a complete classification of multipliers is given, followed by a classification of the conserved density and spatial flux of the conserved current.